As the generalized Proca theory is able to alleviate the \(H_{0}\)  tension at the background level [1]},
our specific choice of the cubic order in terms of the vector field self-interactions in the Lagrangian
also prefers a larger value of \(H_{0}\) ,
determined to be \(H_{0}^{Proca}=71.80^{+1.07}_{-0.72}\)  (\(72.48^{+0.72}_{-0.60}\) )
\(\rm km\;s^{-1}Mpc^{-1}\)  and \(H_{0}^{\Lambda CDM}=66.75^{+1.52}_{-0.73}\)  (\(69.13\pm 0.57\) )
\(\rm km\;s^{-1}Mpc^{-1}\)  at \(68 \%\)  C.L. with CMB (CMB+HST).
The increased value of \(H_0\)  in our model matches the local measurements of HST, SH0ES, H0LiCOW and Megamaser, resulting in that the \(H_{0}\)  tension is reduced to \(\sim 1 \sigma \)  by comparing with the HST data. Furthermore, we have plotted
the evolution of \(H(z)\)  with the initial conditions given by the best-fit from the global fitting results, and found that \(H(z)_{Proca}> H(z)_{\Lambda CDM}\)  at the low redshift with \(z \lesssim 0.55\) .
